FRANCESCO COSTANTINO, FRANCOIS GU ´ERITAUD AND ROLAND VAN DER VEEN
Abstract. We formulate a generalization of the volume conjecture for planar graphs. Denoting by hΓ, ciUthe Kauffman bracket of the graph Γ whose edges are decorated by real “colors” c, the conjecture states that, under suitable conditions, certain evaluations of hΓ, bkcciUgrow exponentially as k → ∞ and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is Γ (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon-Schulten recursion for the quantum 6j-symbols. Assuming that hΓ, bkcciU does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schl¨afli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.
Contents
1. Introduction and basic definitions 1
1.1. The Volume Conjecture 1
1.2. Framework, notation, and results 2
2. Geometry of hyperideal hyperbolic polyhedra 6
2.1. Theorem 1.1–a: Geometric equations associated to faces of P 6 2.2. Theorem 1.1–b: Rigidity of functions satisfying face equations. 7
3. Shadow-state formulation of hΓ, ci 8
4. Recursion relations associated to faces of trivalent graphs 10
5. Proof of Theorem 1.6 14
5.1. Asymptotical behavior of recursion coefficients 14
5.2. Proof of Theorem 1.6 16
Appendix A. Proof of Theorem 1.4 17
A.1. Volumes of hyperideal hyperbolic tetrahedra 17
A.2. Proof of Theorem 1.4 18
References 21
1. Introduction and basic definitions
1.1. The Volume Conjecture. The Volume Conjecture, initially formulated by R. Kashaev [9] and then recast in terms of evaluations of colored Jones polynomials by Murakami and Murakami [13], states that if k ⊂ S3is a hyperbolic knot, and if Jn(A) ∈ Z(A±1) is the nth
colored Jones polynomial of the knot, normalized so that its value on the unknot is 1, then
the following holds: lim n→∞ 2π n log Jn expiπ 2n = Vol(S3\ k). Here i ∈ C is a square root of −1, chosen once and for all.
The conjecture in the above form has been formally checked for the Figure Eight knot [13], for torus knots [10] and their Whitehead doubles [25]. It was also generalized to links and verified for the Borromean link [14] and later for infinite families of links and knotted graphs in [23, 24]. Moreover, there is experimental evidence of its validity for the knots 63, 89 and 820[15]. In [4], an extension of the conjecture was proposed to include the case
of links in connected sums of S2× S1; this extension was proved for the infinite family of
“fundamental shadow links” contained in connected sums of copies of S2× S1. In [6], a
further extension of the conjecture was proposed for links in arbitrary manifolds. Recently another extension was proposed in [16].
One of the main difficulties of the conjecture is that in general it is difficult to understand in sufficient detail the rigid geometric structure of the knot complement. To avoid this problem, the first author formulated [2] a version of the conjecture for planar trivalent graphs and outlined a proof of the conjecture for an infinite class of cases. The intersection of the family of planar graphs with that of knots consists of the unknot, so the new conjecture is actually a generalization in a new direction, in which the topology of the complement of the graphs is easy (it is always a handlebody) and the geometry is rich (polyhedra can deform) without being inaccessible. Indeed we show that under suitable combinatorial conditions the asymptotical behavior of the invariants we consider is related to the hyperbolic volume of the polyhedra whose one-skeleton is described by the graphs. Since a graph has more than one edge, colored Jones polynomials are no longer sufficient to describe the full deformation space of these polyhedra; so our conjecture deals with the natural generalization of these invariants to graphs, namely the Kauffman brackets (also known as quantum spin networks). 1.2. Framework, notation, and results.
1.2.1. On hyperideal hyperbolic polyhedra. Let K3 ⊂ R3 be the open ball representing the
hyperbolic 3-space via the Klein model. Let P ⊂ R3be a finite convex Euclidean polyhedron
such that each vertex wi of P lies in R3\ K3 and each edge of P intersects ∂K3. Let Ci
be the cone from wi tangent to ∂K3 and πi be the half-space not containing wi such that
∂πi∩ ∂K3 = Ci∩ ∂K3. In all the paper we will assume that the 1-skeleton of P, denoted
P(1), is a trivalent graph.
Definition (Truncated hyperbolic hyperideal polyhedra). Given P as above the associated truncated hyperideal hyperbolic polyhedron is Ptrunc:= P ∩T
iπi (see Figure 1). We will
say that a planar graph Γ ⊂ S2 is the 1-skeleton of P (or, with an abuse of notation, of Ptrunc), if there is a homeomorphism mapping (S2, Γ) into (P(2), P(1)).
(Observe that (Ptrunc)(1)is homeomorphic to the graph obtained by replacing each vertex
of P(1) by a triangle as follows: .)
A geometric structure on a truncated hyperideal polyhedron Ptruncis uniquely identified by the exterior dihedral angles at the edges of P hence, if Γ = (Ptrunc)(1), by a map γ : E(Γ) → (0, π], where E(Γ) is the set of edges of Γ. We denote A(P) the set of all possible such angle structures; by a theorem of Bao and Bonahon [1] (see Section 2 for more details), A(P) is a subset of (0, π]E cut out by a finite list of linear inequalities. (We are including the dihedral angle π which corresponds to the case when an edge of P is tangent
Figure 1. An example of Ptrunc(in this case, in red, a truncated tetrahe-dron): remark that Ptruncis compact unless one of the edges of P is tangent to ∂K3.
to ∂K3, in which case the corresponding edge in Ptruncreduces to an ideal point.) To specify
a geometric structure on Ptruncwe will write Ptrunc(γ) for some γ ∈ A(P).
If each edge of P intersects ∂K3 in 2 points (equivalently if γ(e) ∈ (0, π) for all edges) it
is easy to check that Ptrunc(γ) is a compact hyperbolic polyhedron whose edges are of two
kinds: those contained in the truncation faces (i.e. in ∪i(∂πi) ∩ P) and the remaining ones
which are contained in the edges of P. The latter edges of Ptrunc(γ) have well-defined finite lenghts `i and exterior dihedral angles given by γi. Edges that are reduced to ideal points
(with γi = π) have length `i = 0. We will denote by Vol : A(P ) → R the function that
associates to each set of dihedral exterior angles γ ∈ A(P) the hyperbolic volume of the cor-responding Ptrunc. It is well known that Vol : A(P) → R is smooth and satisfies a differential equation called the Schl¨afli formula, stating that d Vol = 1
2 P i`idγi. In particular, (1) `i= 2 ∂Vol(γ) ∂γi . Here is a paraphrase of the result we prove in Section 2.1:
Theorem 1.1. Let Ptrunc be a truncated hyperideal polyhedron.
a. For each face of Ptrunc there is an SL
2(R)-valued equation satisfied by Vol : A(P) →
R of the form: (2) Y i ai(γ) bi(γ) ci(γ) di(γ) cosh(`i 2) sinh( `i 2) sinh(`i 2) cosh( `i 2) = −Id
where i runs over all the edges of P contained in the face, `i= 2 ∂Vol(γ)
∂γi are the edge
lengths of the interior edges of Ptrunc and a
i, bi, ci, di are explicit smooth functions
of the dihedral angles of P adjacent to the face.
b. Moreover, if f : U → R is a function of class C1 defined on an open connected
subset U of A(P), that is close enough to Vol in the C1 sense and satisfies all the
SL2(R)-valued equations (2) associated to the faces of P, then f − Vol is a constant.
We stress that the definition of these equations is purely geometric and does not depend on any quantum object. The equations as in (2) can be seen via (1) as polynomial equations relating the exponentiated dihedral angles and exponentiated edge lengths of Ptrunc, or
alternatively, as partial differential equations satisfied by the volume function.
1.2.2. Invariants of framed graphs, statement of the conjecture and the main result. We turn now to the statement of the volume conjecture for polyhedra.
Definition (KTG). A Knotted T rivalent Graph (KTG) is a finite trivalent graph Γ ⊂ S3 equipped with a “framing”, i.e. an oriented surface retracting to Γ (seen up to isotopy fixing Γ). We denote by V the set of vertices of Γ and E the set of edges.
Definition 1.2 (Admissible coloring). A coloring of Γ is a map c : E → N
2 (whose values
are called colors); it is admissible if for all v ∈ V , if ei, ej, ek∈ E are the edges touching v,
the following conditions are satisfied:
a. c(ei) + c(ej) ≥ c(ek), c(ej) + c(ek) ≥ c(ei), c(ek) + c(ei) ≥ c(ej)
b. c(ei) + c(ej) + c(ek) ∈ N.
As we shall recall in Section 3, given a KTG Γ equipped with a coloring c there is an associated invariant of isotopy of (Γ, c), called the unitary Kauffman bracket (or quantum spin network), denoted hΓ, ciU, valued in the space Hol of functions which are holomorphic on the interior of the unit disc D ⊂ C and whose square is the restriction of a meromorphic function on C (see Remark 3.7).
Definition (Real coloring). Given a sequence of colorings (cn)n∈N on Γ we define the real
coloring γ as the map γ : E → R given by γ(e) = limn→∞cnn(e). We will always tacitly
assume that these limits exist for all e ∈ E, for all the sequences we shall consider. We will also let γ : E → R be defined in terms of γ as
γ(e) = 2π(1 − γ(e)), ∀e ∈ E.
Remark. Clearly γ satisfies weak triangular inequalities as in Definition 1.2–a. Conversely, given a map γ : E → R satisfying strict triangular inequalities around vertices, one can produce a sequence (cn(γ))n∈N of admissible colorings of Γ whose limit is γ by setting e.g.
(cn(γ))(e) := bγ(e)nc. Here and below we take bxc to be the greatest integer less than or
equal to x.
Definition (Evaluation). Let f be the restriction to D of a meromorphic function on C; we denote by evn(f ) the first nonzero coefficient of a Laurent series expansion of f around
A = exp(2nπi).
More generally if f is a holomorphic function on D such that f2extends meromorphically
to C, then we let evn(f ) := ±pevn(f2) (in what follows only the modulus of evn will be
relevant to our computations).
Conjecture 1.3 (Volume Conjecture for polyhedra). Let Γ = P(1) and let γ : E → (0, π]
be the exterior dihedral angles of P. Let also γ := 1 −2πγ and (cn(γ))n∈N be the sequence of
integer colorings defined by cn(γ) = bnγc. Then the following limit holds:
lim n→∞ π nlog evnhΓ, cn(γ)iU = Vol(Ptrunc(γ)).
The factor π instead of the usual 2π in the original volume conjecture can be explained by viewing the polyhedron als one half of the geometric decomposition of the complement of the planar graph. In the special case γ = π the above agrees with the conjecture in [24]. The following result, announced by the first author in [2], can be deduced from the results in [3] but we provide a proof in the Appendix for the sake of self-containedness:
Theorem 1.4. Let Γ be a planar graph obtained from a tetrahedron by applying any finite sequence of the local moves . Then Γ is the 1-skeleton of a hyperbolic ideal polyhedron PΓ and for each γ ∈ A(PΓ) Conjecture 1.3 holds true.
Unfortunately there are plenty of trivalent graphs which cannot be obtained as above from a tetrahedron: e.g. the 1-skeleton of a cube. The main result of this paper is to provide evidence for Conjecture 1.3 in these other cases. Let Γ = P(1) and γ ∈ A(P); let also (cn(γ))n∈N be a sequence of colorings on Γ with limn→∞ cnn(γ) = γ = 1 −2πγ . From now on
we will make the following assumption:
Assumption 1.5. There exists a function f : A(P) → R of class C1 such that for each
map δ : E(Γ) → {−12, 0,12} satisfying the second condition of Definition 1.2 the following limits hold:
lim
n→∞A→explim(iπ 2n) hΓ, cn(γ) + δiU hΓ, cn(γ)iU = exp X ei∈E(Γ) 2δ(ei) ∂f ∂γi .
The limit can be seen as a form of C1 convergence of the growth rate of the evaluations
to the function f . Indeed, if evn(hΓ, cn(γ)i) ∼ exp(nf (γ)) for some regular function f :
A(P) → R and if the asymptotic approximation is itself regular enough, then the above limit holds. (So, although exponential growth does not formally imply Assumption 1.5, it does under suitable regularity hypotheses.)
So our main result below roughly says that if the growth of evaluations of quantum spin networks with underlying graph Γ is exponential and well-behaved (Assumption 1.5), then the growth rate resembles the volume. More precisely we have:
Theorem 1.6. Let Γ, P, γ and cn(γ) be as above and let f be the function whose existence
is supposed in Assumption 1.5. Letting `i = 2∂γ∂f
i, then for each face of P equation (2) is
satisfied.
By Theorem 1.1–b, this is a strong indication towards the fact that the growth rate function f differs from the volume by at most a constant. Of course one of the main questions left open by this work is whether Assumption 1.5 does hold, or at least under which geometric or topological conditions it does.
The proof of Theorem 1.6 is based on the construction of a set of explicit recursion relations which are satisfied by the invariants hΓ, cni. The existence of these recursions can
be proved in general via the techniques used in [8], but it is typically very hard to compute it. We show that for planar graphs there is an easy way of producing such recursions (see Proposition 4.3); in the special case when Γ = our recursions recover the well known Gordon-Schulten recursion for 6j-symbols of Uq(sl2). Then we prove Theorem 1.6 by
studying the asymptotical behavior of the coefficients of these equations, which, surprisingly enough, turn out to be strictly related to those of the geometric Equations (2).
1.2.3. Plan of the paper. Section 2 collects facts about hyperbolic polyhedra, including The-orem 1.1. Section 3 contains the definitions and main properties of the quantum invariants hΓ, ciU. The recursion relations satisfied by hΓ, ciU are shown in Section 4; these are used
in Section 5 to prove Theorem 1.6. Finally in Appendix A we prove Theorem 1.4.
1.2.4. Acknowledgements. F.C. and F.G. were supported by the Agence Nationale de la Recherche through the projects QuantumG&T (ANR-08-JCJC-0114-01) and DiscGroup (ANR-11-BS01-013), ETTT (ANR-09-BLAN-0116-01) respectively. F.G. was also supported by the Labex CEMPI (ANR-11-LABX-0007-01). RvV thanks the Netherlands organization for scientific research (NWO).
2. Geometry of hyperideal hyperbolic polyhedra
Let P and Ptrunc be as in Subsection 1.2.1 and let P∗ be the polyhedron dual to P and E∗ = E(P∗) be its set of edges which is naturally in correspondence with E(P); in what follows we will implicitly use this correspondence to translate a map γ : E(P) → (0, π] to one γ∗ : E(P∗) → (0, π]. Let Γ be a planar graph and Γ∗ be its dual. A result of Steinitz shows that Γ can be the one-skeleton of a convex polyhedron in R3 iff Γ∗is 3-connected i.e.
it cannot be disconnected by deleting 0, 1 or 2 vertices. So we shall suppose from now on that Γ is 3-connected.
The following theorem, proved by X. Bao and F. Bonahon (Theorem 1 of [1], see also [19]), allows one to identify the set A(P) of possible angle structures on Ptrunc:
Theorem 2.1. A map γ : E(P) → (0, π] is the set of exterior dihedral angles on a hyperbolic hyperideal polyhedron combinatorially equivalent to P iff each simple closed curve c in the 1-skeleton of P∗ satisfiesP
e∈cγ
∗(e) > 2π, and each path c0 composed of edges of P∗ such that
∂c0 is contained in the closure of a single face of P∗ but c0 is not, satisfiesP
e∈c0γ∗(e) > π,
where the sums run over the edges of P∗ contained in c (resp. c0). Such a map γ identifies Ptrunc uniquely up to isometry.
Example 2.2 (The hyperideal tetrahedron). If P is a tetrahedron, then so is P∗ and if the set of exterior dihedral angles on P is (α, β, γ) (on three edges adjacent to a vertex of P) and (α0, β0, γ0) on the respective opposite edges, then the conditions for closed curves provided in Theorem 2.1 impose the following inequalities:
α + β + γ > 2π, α0+ β + γ0> 2π, α0+ β0+ γ > 2π, α + β0+ γ0> 2π, (3)
α0+ α + β0+ β > 2π, α0+ α + γ + γ0 > 2π, β0+ β + γ + γ0> 2π. (4)
But since all the angles are in (0, π], Condition (3) entails that the angles on any two adjacent edges sum to more than π, which implies both (4) and the condition on non-closed paths in Theorem 2.1. One also has α + β − γ > 0 (and similarly for all triples of angles around a vertex).
2.1. Theorem 1.1–a: Geometric equations associated to faces of P. Let F ⊂ H2⊂
H3 be a compact, convex polygon with 2p edges and whose exterior angles are all π2. Let
v0, v1, . . . v2p= v0 be its vertices and let `j,j+1= length(vjvj+1); let C(−−−−→vjvj+1) := C(`j,j+1)
where: C(`) :=cosh ` 2 sinh ` 2 sinh`2 cosh`2 , and let M := √1 2 1 −1 1 1 . Proposition 2.3. The following holds:
2p−1
Y
j=0
M C(−−−−→vjvj+1) := M C(−v−−−−→2p−1v0)M C(−−−−−−−→v2p−2v2p−1) · · · M C(−−−→v0, v1) = −Id.
Proof. In the upper-half plane model of H2, up to isometry, we may suppose that v
0= i and
that the edge v0v1 is on the geodesic L connecting −1 and 1, that its points have negative
real part and finally that all the other vertices vi of the polygon F are contained in the
unit disc. The isometry represented by C(−−→v0v1) slides the edge along L until v1 = i, then
the matrix M applies a −π2 rotation with center i. This isometry has then moved v1 in the
same position as the initial position of v0and we can iterate the process until v0returns to
its initial position. The identity expresses the fact that these isometries overall describe the identity map, the minus sign is due to the fact that we are working in SL2(R) thus a full
Let us recall the second law of cosines for hyperbolic triangles:
Proposition 2.4 (Second law of cosines). Let γi, i = 1, 2, 3 be the exterior angles of a
hyperbolic triangle and `i be the lengths of the edges opposite to γi. If {i, j, k} = {1, 2, 3},
then
cosh(`i) =
cos(γj) cos(γk) − cos(γi)
sin(γj) sin(γk)
.
Let now P be a convex trivalent polyhedron in R3 whose truncation gives a
hyper-bolic hyperideal polyhedron Ptrunc, and let {v
s} be the vertices of Ptrunc. Let λtrunc =
v0, v1, · · · , v2p = v0 be the boundary of a face of Ptrunc. To each edge −−−−→vsvs+1 of λtrunc we
may associate the hyperbolic length `s:= `(−−−−→vsvs+1) ∈ R+of −−−−→vsvs+1and an external dihedral
angle γs:= γ(−−−−→vsvs+1) ∈ (0, π] which is defined as π2 if −−−−→vsvs+1is a truncation edge (i.e. if s is
odd) and else it is the external dihedral angle at the edge vsvs+1of Ptrunc. Furthermore, for
each truncation edge (odd s) we may define an “opposite angle” γs0 as the external dihedral angle opposite to vsvs+1 in the truncation triangle containing vsvs+1.
Then the following holds (finishing 1.1–a):
Theorem 2.5. For each loop λ ⊂ P(1) which is the boundary of an oriented face, p−1
Y
j=0
M C
arccoshcos(γ2j+2) cos(γ2j) − cos(γ
0 2j+1)
sin(γ2j) sin(γ2j+2)
M C(`2j,2j+1) = −Id.
Proof. It is a direct consequence of Proposition 2.3, once one computes the length of −−−−−−−→
v2j+1v2j+2via Proposition 2.4.
2.2. Theorem 1.1–b: Rigidity of functions satisfying face equations. Let Γ be the one-skeleton of a polyhedron P and f : U → R be a smooth function defined on a connected open subset U of A(P). We say that f satisfies the matrix-valued face equations if, setting `i := 2∂γ∂fi, the SL2(R)-valued equations described in Theorem 2.5 are satisfied
for all simple closed curves λtrunc in the 1-skeleton of Ptrunc formed by the boundaries of
the non-truncation faces of Ptrunc. The Schl¨afli formula (1) implies that Vol, in particular,
satisfies the matrix-valued face equations. The goal of this subsection is to check that any function f close to Vol in the C1 sense that also satisfies the matrix-valued face equations, is equal to Vol up to an additive constant.
This is a simple consequence of the rigidity of hyperideal polyhedra (Theorem 2.1). Indeed, if the partial derivatives of f were different from those of Vol at a given point γ ∈ (0, π]E(P), then f would define a small deformation Pf of the hyperideal polyhedron P
(with the same angles γ but different edge lengths), which is impossible. More precisely, any function f as above allows us to define an (H3
, Isom(H3))-structure
on a neighborhood of the polyhedron Ptrunc⊂ R3 with dihedral angles γ, as follows. First,
let U1and U2be tubular neighborhoods of the 1- and 2-skeleta of Ptruncin R3, respectively.
Endow Ptrunc with formal (real) edge lengths, given by `i = 2∂γ∂fi at interior edges, and
by Proposition 2.4 (independently of f ) at truncation edges. Let fU1 denote the universal
cover of U1. The above length data is enough to define a developing map eΦf : fU1 → H3
respecting all angles, edge lengths and framings, in the sense that eΦf takes the lift of the
loop around a 2p-gonal face F of Ptruncto a path Ff formed of 2p coplanar segments in H3,
each orthogonal to the next. We may assume that Ff lies in H2⊂ H3.
The identity of Proposition 2.3, associated to the loop around the p-gonal face of P containing F , implies that Ff closes up in H2, with the given edge lengths, to form a
lengths, it follows that Ff bounds a convex right-angled 2p-gon of H2. As for truncation
triangles, they similarly close up under eΦf since eΦf assigns them the same angles and edge
lengths as truncation triangles of Ptrunc. Therefore eΦ
f decends to a map Φf : U1 → H3
which can be extended to the universal cover of U2. As U2is simply connected, we actually
have a developing map Φf : U2→ H3. Since Φf and ΦVol are very close, it follows that Φf
(like ΦVol) defines a realization of ∂Ptruncas the boundary of a convex truncated hyperideal
polyhedron Ptrunc
f . By rigidity of convex polyhedra, Pftruncand Ptruncare actually isometric
(via Φf) and have in particular the same edge lengths, which means that f and Vol have
the same partial derivatives at γ, for all real tuples γ. Hence f − Vol is locally constant: Theorem 1.1–b is proved.
3. Shadow-state formulation of hΓ, ci
In this section we will recall the definition of quantum spin networks through their “shadow-state formulas” (as these formulas will be used later on). We now fix a diagram D of Γ such that the blackboard framing coincides with that of Γ (it is easy to check that it exists), and an admissible coloring c on Γ, and recall how the Kauffmann bracket hΓ, ci is defined. Let A ∈ C, q = A2, {n} := A2n− A−2n, [n] := {n}/{1} , [n]! := n Y j=1 [j] , [0]! := 1 , n k := [n]! [k]![n − k]!
and similarly for multinomials. In order to be able later on to coherently choose branches of square roots, we define ∆Hol(A) to be the set of functions which are holomorphic inside the unit disk D and whose square extends to a meromorphic function on C. We define p[k] ∈ ∆Hol(A) as the root of [k] which at A = 1 is√k and then we extend this choice on products of quantum integers so thatp[n]! := Qnj=1p[j] and q[n]!1 := √1
[n]!. (Remark
that these square roots exist on D because [k] = A2−2k(Pk−1
i=0 A
4i) and both terms admit a
square root on D).
Definition (Unknot). We define the value of the colored 0-framed unknot as follows: a := (−1)2a[2a + 1]. Let also ∆(a, b, c) := s [a + b − c]![b + c − a]![c + a − b]! [a + b + c + 1]! .
The (unitary normalization of the) tetrahedron or 6j-symbol is then defined as follows: Definition 3.1 (The tetrahedron or symmetric 6j-symbol). Let us define the value of the admissibly colored tetrahedron embedded as a planar graph in R2 as:
a b c d f e !U
:= i2(a+b+c+d+e+f )∆(a, b, c)∆(a, e, f )∆(d, b, f )∆(d, e, c) (5) × k=min Qj X k=max Ti (−1)k k + 1 k − T1, k − T2, k − T3, k − T4, Q1− k, Q2− k, Q3− k, 1
where T1= a + b + c, T2= a + e + f, T3= d + b + f, T4= d + e + c, Q1= a + b + d + e, Q2=
a + c + d + f, Q3= b + c + e + f .
Definition 3.2 (The crossed tetrahedron). Let us define the value of the admissibly colored tetrahedron, embedded in R3 as in the diagram below, by:
a b c d f e := i2(e+b−a−d)
A2(e2+e+b2+b−a2−a−d2−d) c ab d
f
e !U
Remark 3.3. If in formula (5) one color, say f , is 0 then the admissibility conditions force e = a and d = b and, using also that a + b + c ∈ Z we get:
a b c d f e !U = i2a+2bp[2a + 1][2b + 1] −1
which does in fact not depend on c.
The preceding formulas can be used as “building blocks” to define invariants of colored KTG’s up to isotopy in R3
: let (Γ, c) be a colored KTG, D ⊂ R2 be a diagram chosen so
that the framing of Γ coincides with the blackboard framing; let V, E be the sets of vertices and edges of Γ, and C, F the sets of crossings and edges of D. Since each edge of D is a sub-arc of one of Γ it inherits a coloring from c. Let the regions r0, . . . , rm of D be the
connected components of R2\ D with r0 the unbounded one; we will denote by R the set
of regions and we will say that a region “contains” an edge of D or a crossing if its closure does.
Remark 3.4. Applying an isotopy to D we can force any region to become the “unbounded” one, denoted r0 above. We will exploit this freedom in the proof of Proposition 4.3.
Definition 3.5 (Shadow-state). A shadow-state s is a map s : R → N
2 such that s(r0) = 0
and whenever two regions ri and rj contain an edge f of D then s(ri), s(rj), c(f ) form an
admissible triple as in Definition 1.2.
Given a shadow-state s, we can define its weight as a product of factors indexed by the local building blocks of D i.e. the regions of D, the vertices of Γ, and the crossings. To define these factors explicitly, in the following we will denote by a, b, c the colors of the edges of Γ (or of D) and by u, v, t, w the values of a shadow-state on the regions (which with an abuse of terminology we will call the shadow-states of the regions).
(1) If r is a region whose shadow-state is u and χ(r) is its Euler characteristic, ws(r) := u χ(r) = (−1) 2u{2u + 1} {1} χ(r)
(2) If v is a vertex of Γ colored by a, b, c and t, u, v are the shadow-states of the regions containing it then ws(v) := a b c t v u
(3) If c is a crossing between two edges of G colored by a, b and u, v, t, w are the shadow-states of the regions surrounding c then
ws(c) := w t a u b v
From now on, to avoid a cumbersome notation, given a shadow-state s we will not explicitly write the colors of the edges of each graph providing the weight of the local building blocks of D as they are completely specified by the states of the regions and the colors of the edges of Γ surrounding the block. Then we may define the weight of the shadow-state s as:
(6) w(s) = Y r∈R χ(r)Y v∈V Y c∈C .
Then, since the set of shadow-states of D is easily seen to be finite, we may define the hΓ, ciU
as
hΓ, ciU:= X
s∈shadow states
w(s).
The following result was first proved by Reshetikhin and Kirillov [18] (see also [5] for a skein theoretical proof) and shows that shadow state-sums provide a different approach to the computation of the so-called “quantum spin networks” or “Kauffman brackets” in their unitary normalizations hΓ, ciU∈ ∆Hol(A):
Theorem 3.6 (Shadow-state formula for Kauffman brackets.). hΓ, ciU is a well defined
invariant up to isotopy of Γ. In particular, if Γ has a planar diagram D with respect to which the framing of Γ is the blackboard framing, then formula (6) reduces to:
(7) hΓ, ciU= X s∈shadow states Y r∈R χ(r) Y v∈V .
Remark 3.7. (1) The above defined invariant coincides with the so-called “unitary quantum spin network” defined and studied via recoupling theory in [11].
(2) For each graph Γ, (hΓ, ciU)2
is the restriction to D of a meromorphic function on C whose poles are in the set {0, exp(iπpq ), p, q ∈ Z}. Indeed a close inspection to
Formula (6) or (7) shows that the only odd powers of square roots appearing in the expression are those associated to the colors of the edges of Γ, and thus can be factored out of the whole sum in (7). (All other square roots arising from 6j-symbols involve one edge color and two state parameters. Each such square root appears twice: once for each endpoint of the edge.)
As an example we state a simple corollary that we will use often below. Its proof is a simple application of the shadow state formula but can also be found in [12].
Corollary 3.8 (Triangle formula). Suppose Γ is a KTG such that three edges bound a flat triangular disk. Then we have (representing Γ on the left hand side):
(8) * a c b f e d +U = * a b c d f e +U * a c b +U .
4. Recursion relations associated to faces of trivalent graphs
In [8], it was proved that the sequence of quantum spin networks associated to a link (better known as its colored Jones polynomials) is holonomic, i.e. satisfies a set of recurrence relations. More explicitly, if k ⊂ S3is a knot, let Jn= hk,n−12 i be the sequence of its colored
Jones polynomials and extend it to a map J : Z → C[A±1] by setting J1= 1, J0= 0, J−n=
−Jn. Let S = {f : Z → C[A±1]} and let L, M : S → S be defined as L(f )(n) = f (n + 1)
a0 a1 b0 a2 b1 b2 b3 b4 b5 a3 a5 a4 D
Figure 2. Some of the notation for Proposition 4.3.
Theorem 4.1. There exists a polynomial QA(L, M ) depending on k such that
QA(L, M )(J ) = 0.
(Actually the above statement is weaker than the original one: we refer to [8] for full details).
Even if the techniques of the above theorem may be adapted to prove similar statements for general colored KTG invariants, for example along the lines of [7], it is difficult to make this explicit in general. In this section, we shall consider the case when Γ is the 1-skeleton of a polyhedron P and, given the embedded cycle ∂D ⊂ Γ formed by the boundary of a face of P , we will provide a family of instances of these recursion polynomials. (Note that, as there are now many “colors” given by the map c : E → N
2, we should also expect many
recursions. Our technique can be easily generalised to any knotted trivalent graph Γ and any cycle in Γ which is the boundary of a disc embedded in S3\ Γ, but we won’t need this level of generality here.)
Let Γ be a KTG which is the 1-skeleton of a polyhedron P. Let c : E → N 2 be an
admissible coloring on the edges of Γ. Let D ⊂ S3\ Γ be an oriented disc isotopic to a face of P. Fix an orientation of D, a basepoint p0∈ ∂D and denote by e0, . . . ep−1, ep = e0 the
edges contained in ∂D (numbered by walking along ∂D in the clockwise direction starting from p0). Let ai, bi, i = 0 . . . p − 1 be the colors respectively of eiand of the edge of Γ sharing
a vertex with ei, ei+1(mod p) (see Figure 2).
Definition 4.2 (Admissible perturbations). An admissible perturbation of the coloring c of Γ around D is a map δ : {e0, . . . ep−1} → {±12} such that the coloring c + δ (defined to
be equal to c on the edges outside ∂D and to be ei 7→ ai+ δ(ei), i = 0, . . . p − 1 on ∂D) is
admissible.
We consider the following proposition as a “quantum version” of Proposition 2.3: Proposition 4.3 (Circle recursions). For any pair (δ(e0), δ(ep−1)) ∈ {±12}2, the following
holds:
(9) X
δ
k(c, δ)hΓ, c + δiU= Z(c, δ(e0), δ(ep−1))hΓ, ciU
where δ ranges over all admissible perturbations of c around D whose value on e0 and ep−1
are respectively δ(e0), δ(ep−1), and the functions Z(c, δ(e0), δ(ep−1)) and k(c, δ) are defined
Define Z(c, δ(e0), δ(ep−1)) = V(p − 1) and k(c, δ) =Q p−2
i=0 V(i) in terms of the quotient V:
(10) V(j) := aj bj aj+1 a0j+1 1 2 a0j a0j bj a0j+1 a0 j+1 0 a0j .
Proof. Let Γ0 = Γ0(δ0, δp−1) be the colored planar graph obtained by modifying Γ around
e0∩ ep−1 as follows: (11) bp−1 a0 ap−1 D −→ bp−1 a0 ap−1 a0p−1 a00 1/2
where in the picture both the framings of Γ and Γ0 are supposed to be the blackboard framings. We will call r0the region bounded by the small triangle created by the move, and
let r00 be its complement in D.
The idea of the proof is to express hΓ0iU in terms of hΓ, c + δiUtwo ways. First directly
using the triangle formula (8). Second by applying the shadow state formula (7) with region r0 having the role of the “unbounded” region and comparing it to the similar state sum for
various colorings of hΓi.
If we multiply (9) by the denominator Z0 of Z then the right hand side becomes a00 a0p−1bp−1 ap−1 a0 1 2 hΓ, ciU= hΓ0iU,
using (8). The proof is finished once we show that the left hand side of the same equation equals the expression of hΓ0iUin terms of the shadow state formula with distinguished region
r0. In other words we compute hΓ0iU by stipulating the triangular region r0 to be the one
carrying shadow state 0. Let RΓ0 be the set of all regions of Γ0. The region r00 is a disc
and its only admissible shadow state is 12. Moreover in a neighborhood of r00we will see one region ri∗∈ RΓ0 for each edge ei (∂ri∗ contains ei, and ri∗∩ r0= ∅ unless i = p − 1 or i = 0,
in which case the intersection is one of the edges of r0, colored respectively by a0p−1 and by
a00).
We have hΓ0iU = P
shadow statesw0(s) where w0(s) is the weight of the shadow state
s : RΓ0 → N 2 for Γ
0. Observe that the only admissible shadow-states of r∗
i are a0i = ai+ δi
with δ admissible (as in the statement). So we may also write: hΓ0iU=X δ X s∈S(δ) w0(s) =X δ w0(δ)
where for each perturbation δ we let S(δ) be the set of all the shadow-states whose values on regions r∗i are a0i = ai+ δi for all i ∈ {0, . . . p − 1} and w0(δ) :=Ps∈S(δ)w
0(s). We claim
that w0(δ) = hΓ, c + δiUk(c, δ)Z0 for all admissible δ, which (by summing over δ) will finish the proof, since Z0 is independent of δ.
Indeed, fix δ. By the shadow-state formula (7) applied to Γ with the region D colored by 0, one sees that hΓ, c + δiU=P
s∈S(δ)w(s) is a state-sum over the same set S(δ) of colorings
(of RΓr {D} ' RΓ0 r {r0, r00}) as hΓ0iU is — the colorings being extended to be 0 on D
instead of 1 2 on r
0
and to the vertices contained in ∂D. More explicitly we claim that for any s ∈ S(δ) one has
w0(s)
w(s) = k(c, δ)Z
0 (independent of s). Indeed the weights differ only by the vertices touching
D and by the presence of the weight of the region r00 (which is −[2]):
w0(s) = −[2] 1 2 a00 a0 a00 1 2 0 1 2 a0p−1 ap−1 a0p−1 1 2 0 0 a0p−1 a0p−1 bp−1 a00 a00 p−2Y i=0 ai bi ai+1 a0i+1 1 2 a0i × [off-D terms] w(s) = a0 0 p−1 a 0 p−1 bp−1 a00 a00 p−2Y i=0 a0i bi a0i+1 a0i+1 0 a0i × [off-D terms].
Taking the ratio (all the symbols represent nonzero functions in ∆Hol(A)) and using Remark 3.3, we get: w0(s) w(s) = −[2] 1 2 a00 a0 a00 1 2 0 1 2 a0p−1 ap−1 a0p−1 1 2 0 0 a0p−1 a0p−1 bp−1 a00 a00 0 a0p−1 a 0 p−1 bp−1 a00 a00 p−2 Y i=0 ai bi ai+1 a0i+1 1 2 a0 i a0i bi a0i+1 a0i+1 0 a0i = Z0k(c, δ)
by definition of Z0 and k(c, δ). Multiplying by w(s) and summing over s ∈ S(δ) gives the
desired identity.
To illustrate our recursions consider the simplest case of a triangular face p = 3. We recover the well-known three-term recursion for the 6j-symbol named after Gordon and Schulten [20]. As the proof shows, the Gordon-Schulten recursion factorizes into two in-stances of the circle recursion, suggesting that the circle recursion is more fundamental. Corollary 4.4 (Gordon-Schulten recursion). Let e0, e1, e2 be a triangular face of Γ and
define the edge coloring 1i by 1i(ej) = δi,j. Also define δ±= 12(10± 11+ 12).
X ± k(c, δ±)k(c + δ±, −δ∓) Z(c + δ±,−12 , −1 2 ) hΓ, c ± 11iU= −Z(c,1 2, 1 2) + X ± k(c, δ±)k(c + δ±, −δ±) Z(c + δ±,−12 , −1 2 ) ! hΓ, ciU
Proof. For ∈ {−1, 1} the only two admissible perturbations δ with δ(e0) = δ(e2) = 2 are
δ±. Hence the circle recursion reads:
X ± k(c, δ±)hΓ, c + δ±iU= Z(c, 2, 2)hΓ, ci U.
Applying the recursion for = −1 to both terms on the left hand side of the recursion for
= 1 yields the result.
The coefficients of the recursion may be written out explicitly using the formulae in the next section but as we make no further use of this example we have not printed them here. See for example [21], Prop. 4.1.1.
5. Proof of Theorem 1.6
Let us fix the notation for all this section. Let Γ be the 1-skeleton of a hyperbolic, hyperideal polyhedron P and let γi : E(P) → (0, π] be the exterior dihedral angles of P. Let
also λ ⊂ Γ be an embedded oriented curve forming the boundary of a face D of P and λtrunc
be the associated curve in Ptrunc, e
0, e1, . . . ep−1 be the sequence of edges of Γ contained in λ
and v0, v1, . . . v2p−2, v2p−1be the vertices of Ptrunccontained in λtruncso that −−−−−→v2iv2i+1⊂ ei.
We will consider a sequence c(n)
of N-valued colorings on Γ such that for each edge, limn→∞c (n)(e i) n = 1 − γi 2π (and so in particular c (n)(e
i) ∈ [n2, n) definitely). Finally we will
let ai:= c(n)(ei) (suppressing the n from the notation with a little abuse) and let bi be the
color of the edge of Γ sharing a vertex with ei and ei+1 with respect to the coloring c(n).
Our goal is to relate the recurrence equations provided by Proposition 4.3 to the geometric equations associated by Proposition 2.3 to λtrunc.
5.1. Asymptotical behavior of recursion coefficients. Let us compute explicitly the values of the coefficients k(c, δ) of Proposition 4.3. Using formula (5), we get that V(j) from (10) is a matrix whose rows and columns are indexed respectively by δ(ej), δ(ej+1) ∈ {±12},
namely: V(j) = ifδj =12 ifδj = −12 ifδj+1= 12 i−1 q[1+a j+aj+1−bj][2+aj+aj+1+bj] [2+2aj][2+2aj+1] i q[1−a j+aj+1+bj][−aj+1+aj+bj] [2aj][2aj+1+2] ifδj+1= −12 i q[1+a
j−aj+1+bj][aj+1−aj+bj] [2aj+2][2aj+1] i
q[a
j+aj+1−bj][aj+aj+1+bj+1] [2aj][2aj+1]
where, as explained in Section 3, we chose the square roots by stipulating that for all k ∈ N, p[k] was√k at A = 1, and so by holomorphicityp[k] is a positive real number on the arc A = exp(iθ) for θ ∈ [0,π
k) and it is a positive real multiple of −i on θ ∈ ( π k,
2π
k). Equivalently,
if A = exp(2niπ) thenp[k] ∈ R+if k < n and it is a positive real multiple of −i if k ∈ (n, 2n).
Let us now compute the asymptotical behavior of the recursion coefficients found in the table above when we evaluate at A = exp(2niπ). By hypothesis there exist angles γj, γj+1, βj∈
[0, π) such that lim n→∞2π(1 − aj n) = γj, limn→∞2π(1 − aj+1 n ) = γj+1, and limn→∞2π(1 − bj n) = βj (and so in particular aj, bj, aj+1∈ [n2, n)). Remark that since these angles satisfy Bonahon
and Bao’s conditions, one has βj+ γj+ γj+1> 2π, and replacing this to compute the square
roots of the evaluation of the above quantum integers we get: evn q [2aj] ∼ evn q [2 + 2aj] ∼ −i s sin(γj) sin(πn), evn q [2aj+1] ∼ evn q [2 + 2aj+1] ∼ −i s sin(γj+1) sin(πn) evn q [aj− aj+1+ bj] ∼ evn q [1 + aj− aj+1+ bj] ∼ s sin(γj 2 − γj+1 2 + βj 2) sin(πn) evn q [1 + aj+ aj+1+ bj] ∼ evn q [2 + aj+ aj+1+ bj] ∼ −i s | sin(γj 2 + γj+1 2 + βj 2)| sin(nπ) ,
where the symbol fn ∼ gnstands for limn→∞fgn
n = 1. Then the matrix V(j) is asymptotically
equivalent to: q | sinγj+γj+1−βj 2 sin γj+γj+1+βj 2 | −i q sinγj−γj+1+βj 2 sin γj+1−γj+βj 2 −i q sinγj−γj+1+βj 2 sin γj+1−γj+βj 2 − q | sinγj+γj+1−βj 2 sin γj+γj+1+βj 2 | psin(γj) sin(γj+1) . A straightforward computation shows that
(12) evn(V(j)) ∼ R S(j) psin(γj) sin(γj+1) R−1U where S(j) = q sin(γj−γj+1+βj 2 ) sin( γj+1−γj+βj 2 ) q − sin (γj+γj+1−βj 2 ) sin( γj+γj+1+βj 2 ) q − sin (γj+γj+1−βj 2 ) sin( γj+γj+1+βj 2 ) q sin(γj−γj+1+βj 2 ) sin( γj+1−γj+βj 2 ) and R :=exp( iπ 4) 0 0 exp(−iπ4) , U := 0 −i −i 0 . Lemma 5.1. The following holds:
S(j) psin(γj) sin(γj+1) = C(`2j+1,2j+2) = cosh(`2j+1,2j+2 2 ) sinh( `2j+1,2j+2 2 ) sinh(`2j+1,2j+2 2 ) cosh( `2j+1,2j+2 2 ) !
where `2j+1,2j+2 is the length of the edge opposite to the angle βj in a hyperbolic triangle
whose exterior dihedral angles are γj, γj+1, βj, or, equivalently, is the length of the edge
−−−−−−−→
v2j+1v2j+2in Ptrunc.
Proof. Remark that, by the inequalities examined in Example 2.2, all the arguments of the square roots in the coefficients of Sj are positive. Moreover a direct computation shows
det(Sj) = sin(γj) sin(γj+1) > 0. So M :=
Sj
√
sin(γj) sin(γj+1)
∈ SL2(R) is a real positive matrix
fixing ±1 ∈ ∂∞H2; hence it represents a hyperbolic translation of a certain length λ ∈ R+, and has the form M = C(λ). Identifying diagonal terms of C(λ) yields
coshλ 2 =
r
sinγj −γj+1+βj2 sinγj+1−γj +βj2
sin γjsin γj+1 hence cosh λ =
cos γjcos γj+1− cos βj
sin γjsin γj+1
. This equals cosh(`2j+1,2i+2) by Proposition 2.4, so λ = `2j+1,2i+2.
Putting Lemma 5.1 and Equation (12) together we get: Proposition 5.2. The following limit holds:
lim
n→∞evn(V(j)) = RC(`2j+1,2j+2)R −1U.
Hence, letting k(cn, δ) be as defined in Proposition 4.3 and denoting by Xδ,δ0 the entry at
row δ and column δ0 of any given matrix X, the following holds: lim n→∞evn(k(cn, δ)) = p−1 Y j=0 RC(`2j+1,2j+2)R−1U δ(ej),δ(ej+1).
Similarly, evaluating the Z(cn, δ(e0), δ(ep−1)) of Proposition 4.3, we get: Z = ifδ(e0) =12 ifδ(e0) = −12 ifδep−1= 1 2 i −1q[1+a0+ap−1−bp−1][2+a0+ap−1+b0] [2+2a0][2+2ap−1] i q[1−a p−1+a0+bp−1][−a0+ap−1+bp−1] [2a0][2ap−1+2] ifδep−1= − 1 2 i q[1+a
p−1−a0+bp−1][a0−ap−1+bj]
[2a0+2][2ap−1] i
q[a
0+ap−1−bp−1][a0+ap−1+b0+1] [2ap−1][2a0]
and arguing exactly as in the preceding case we get the following: Lemma 5.3. We have the following asymptotic behavior:
lim n→∞evn(Z(cn, δ(e0), δ(ep−1))) = R cosh(`p−1,0 2 ) sinh( `p−1,0 2 ) sinh(`p−1,0 2 ) cosh( `p−1,0 2 ) ! R−1U
where `p−1,0 is the length of the edge opposite to the angle βp−1 in the hyperbolic triangle
whose exterior dihedral angles are γ0, γp−1, βp−1.
5.2. Proof of Theorem 1.6. To each edge of λtrunc we associate its length `
j,j+1 :=
`(−−−−→vjvj+1), ∀j ∈ {0, 1, . . . 2p − 1} and, as in Section 2.1, let :
C(−−−−→vjvj+1) := cosh`j,j+1 2 sinh `j,j+1 2 sinh`j,j+1 2 cosh `j,j+1 2 ! , M := √1 2 1 −1 1 1 .
Furthermore, suppose that Assumption 1 holds for the sequence hΓ, cniU and let f :
A(P) → C be the function of class C1 whose existence is postulated by the assumption.
Then for each j ∈ {0, . . . , p − 1} we associate to the segment −−−−−→v2jv2j+1 the “asymptotic
quantum length” 2∂γ∂f
j and a diagonal matrix as follows:
Q(−−−−−→v2jv2j+1) = ∂f ∂γj 0 0 −∂f ∂γj ! .
Theorem 5.4. The function f satisfies the following matrix-valued differential equation:
p−1
Y
j=0
M C(−−−−−−−→v2j+1v2j+2)M Q(−−−−−→v2jv2j+1) = −Id
Moreover if V (γ) : A(P) → R is the hyperbolic volume of Ptrunc then V also satisfies the
above equations.
Proof. Fix values δ(e0), δ(ep−1) ∈ {±12}. By Proposition 4.3 the following recursion relations
are satisfied:
X
δ
k(cn, δ)hΓ, cn+ δiU= Z(cn, δ(e0), δ(ep−1))hΓ, cniU
where the sum is taken over all the admissible perturbations δ of cn around the face D of P
bounded by λ and whose values on e0 and ep−1 are δ(e0), δ(ep−1) (see Definition 4.2). By
letting δ(e0) and δ(ep−1) range over {±12} the above equation can be viewed as a single,
2 × 2-matrix valued equation. By Assumption 1.5, if we divide the equation by hΓ, cniU,
evaluate at A = exp(2niπ) (using Lemma 5.1) and take the limit n → ∞, then we get: X δ lim n→∞evn(k(cn, δ)) exp (2δi ∂f ∂γi ) = lim n→∞evn(Z(cn, δ(e0), δ(ep−1)).
Using Proposition 5.2 we realize that the above equality can be expressed as:
Q(−−−−−−−→v2p−2v2p−1) p−2
Y
i=0
RC(−−−−−−−→v2i+1v2i+2)R−1U Q(−−−−−→v2iv2i+1) = RC(−v−−−−→2p−1v0)R−1U.
(13)
Now observe that a straightforward computation shows the following: R−1U Q(−−−−−→v2iv2i+1)R = M C(−−−−−→v2iv2i+1)M.
Replacing this in (13) and simplifying we get
M C(−−−−−−−→v2p−2v2p−1)M p−2
Y
i=0
C(−−−−−−−→v2i+1v2i+2)M C(−−−−−→v2iv2i+1)M = R−1U RC(−v−−−−→2p−1v0)R−1U R.
Finally a last computation shows that
R−1U RC(−v−−−−→2p−1v0)R−1U R = − (C(−v−−−−→2p−1v0)) −1
and thus we conclude by multiplying on the right by C(−v−−−−→2p−1v0). The fact that the same
equation is satisfied by Vol(Ptrunc) is a consequence of the Schl¨afli formula and of Theorem
2.5.
Appendix A. Proof of Theorem 1.4
A.1. Volumes of hyperideal hyperbolic tetrahedra. In [17] J. Murakami and M. Yano found a formula for the hyperbolic volume of a hyperbolic compact tetrahedron and the formula was later shown by A. Ushijima to hold also for truncated tetrahedra (i.e. the compact polyhedra obtained by truncating hyperideal tetrahedra as explained in Subsection 1.2). We now recall this formula.
With the notation of Example 2.2 for the dihedral angles of a tetrahedron, let A, B, C and A0, B0, C0be respectively − exp(−iα), − exp(−iβ), − exp(−iγ) and − exp(−iα0), − exp(−iβ0), − exp(−iγ0) 1. Let Li 2(z) := Rz 0 log(1−t) −t dt = P n>0 zn
n2 be the dilogarithm function
(well-defined on C \ [1, ∞)) and Λ(x) :=Rx
0 − log |2 sin t| dt. Define
U (z) := 1
2 Li2(z) + Li2(zABA
0B0) + Li
2(zACA0C0) + Li2(zBCB0C0)
(14)
−Li2(−zABC) − Li2(−zAB0C0) − Li2(−zA0BC0) − Li2(−zA0B0C)
∆(x, y, z) := −1 4 Li2(− xy z ) + Li2(− yz x) + Li2(− zx y ) (15) +Li2(− 1 xyz) + (log(x)) 2+ (log(y))2+ (log(z))2
V (z) := ∆(A, B, C) + ∆(A, B0, C0) + ∆(A0, B, C0) + ∆(A0, B0, C) +1
2(log(A) log(A
0) + log(B) log(B0) + log(C) log(C0)) + U (z).
1In [17] internal dihedral angles are used, this accounts for the change in the definition of
For later purposes, recall also that for all θ ∈ (0, 2π) one has =(Li2(exp iθ)
2 ) = Λ( θ 2), so
replacing the values of A, B, C, A0, B0, C0 and setting z = exp(is) in (14) we get =(U (exp(is))) = Λs 2 + Λ s − (α + α 0+ β + β0) 2 (16) +Λ s − (α + α 0+ γ + γ0) 2 + Λ s − (γ + γ 0+ β + β0) 2 − Λ s − (α + β + γ) 2 −Λ s − (α + β 0+ γ0) 2 − Λ s − (α 0+ β + γ0) 2 − Λ s − (α 0+ β0+ γ) 2 . Let now z± be the two nontrivial solutions of the equation (which reduces to a degree 2
polynomial equation):
d
dzU (z) ∈ πi
zZ. As shown by Ushijima [22] these can be expressed as:
(17) z±= −2
sin(α) sin(α0)+sin(β) sin(β0)+sin(γ) sin(γ0)±√det(G) AA0+BB0+CC0+ABC0+A0BC+AB0C+A0B0C0+ABCA0B0C0
where (18) G =
1 cos(α) cos(β) cos(γ0)
cos(α) 1 cos(γ) cos(β0) cos(β) cos(γ) 1 cos(α0) cos(γ0) cos(β0) cos(α0) 1
Then the following was first proved by Murakami and Yano in [17] for compact hyperbolic tetrahedra and was later shown by A. Ushijima [22] to hold for the case of a truncated hyperideal tetrahedron:
Theorem A.1 ([17],[22]). The volume of the truncated hyperbolic tetrahedron whose exterior dihedral angles are as in Example 2.2, is
Vol(T et) = = (V (z−)) = −= (V (z+)) = =
U (z−) − U (z+)
2
. A.2. Proof of Theorem 1.4. We will need the following:
Lemma A.2. Let α ∈ (0, 1) and let (an)n∈Nbe a sequence of integers such that limn→∞ ann =
α. Then {an} is meromorphic, for n sufficiently large it has no pole at exp 2niπ and the
following holds: lim n→∞ π nlog evn({an}!) ian = −Λ(πα) where we imply that the argument of the log is real positive.
Similarly, if α ∈ (1, 2) and (an)n∈N is a sequence of half-integers such that limn→∞ ann =
α, then {an}! has a simple zero at exp 2niπ for n big enough and the following holds:
lim
n→∞
π nlog
evn({an}!)
(−1)n+an2n2ian+1exp(−iπ 2n)
!
= −Λ(πα)
Proof. We limit ourselves to a sketch, see [3] for a detailed proof. Clearly evn(f g) =
evn(f )evn(g), and if k is not a multiple of n, evn({k}) = 2i sin(πkn ) while evn({n}) =
−2n exp(−iπ2n ) and thus if an ∈ [0, n) (which is true for the first statement),
evn({an}!) = an Y k=1 2i sinπk n ∼ i anexp−n Λ( πan n ) − Λ( π n) π ; while if an ∈ [n, 2n), evn({an}!) = n−1 Y k=1 2i sinπk n ! −2n exp−iπ 2n an Y k=n+1 2i sinπk n ! =
−2in−1n2exp−iπ
2n an Y k=n+1 −i 2 sinπk n ∼ −(−i)an+1(−1)n2n2exp−iπ
2n exp −n Λ(πan n ) − Λ(π + π n) π . We can now prove Theorem 1.4. For the sake of self-containedness we start by sketching the proof of the following proved in [3] for the skein normalization of the tetrahedron: Theorem A.3. Let Γ be the 1-skeleton of a hyperideal tetrahedron T et whose exterior dihedral angles are as in Example 2.2 and let (cn)n∈N be a sequence of colorings on Γ such
that cn∈ [n2, n) and limn→∞2π(1 −cnn) equals the corresponding exterior dihedral angles of
T et. Then, lim n→∞ π nlog |evnhΓ, cni U| = Vol(T ettrunc).
and Conjecture 1.3 holds in this case.
Proof. The idea of the proof is to first identify the leading term in (5), then to recognize it as the volume of the tetrahedron using Theorem A.1. Despite the signs present in (5), there will be no cancellation. Let us start by computing the evaluation at A = exp(2niπ) of ∆(a, b, c) (as in (5)) using Lemma A.2:
lim n→∞ π nlog(|evn( p ∆(a, b, c))|) =1 2 Λ(α + β − γ 2 )+ Λ(α − β + γ 2 ) + Λ( −α + β + γ 2 ) − Λ( α + β + γ 2 )
and similarly for ∆(a, e, f ), ∆(d, b, f ), ∆(d, e, c). Indeed for instance we have limn→∞ann =
1 −2πα (and similarly for the other ratios) and so lim n→∞ π nlog(evn([an+ bc− cn]!)) = −Λ(π − α + β − γ 2 ) = Λ( α + β − γ 2 ).
Observe that the above formula equals the imaginary part of Equation (15) (recall that for all θ ∈ (0, 2π) one has =(Li2(exp iθ)
2 ) = Λ( θ 2)).
Now let us concentrate on the summation in Formula (5). Let Sk be the kth summand
and remark that k ranges in [max{Ti}, min{Qj}]. We claim that Sk has a simple zero at
A = exp 2niπ if k ∈ [max{Ti}, n − 2] and a zero of higher order otherwise, so only the
summands Sk with k ∈ [max{Ti}, n − 2] need be considered for the purpose of computing
Indeed by the inequalities of Example 3.1 translated in terms of γ ∼ 2π(1 −cn
n), for n
large enough, using the notation of Definition 3.1, one has n < max({Ti}) < 2n; moreover
by hypothesis c(ei) ∈ [n2, n) for every ei∈ E(Γ) so c(ei)+c(ej)−c(ek) < n for every 3-uple of
edges touching a common vertex (in whatever order). Since the summation index k ranges from max({Ti}) to min({Qj}) the differences k − Tiand Qj− k are bounded above by a term
of the form c(ei) + c(ej) − c(ek) < n (for some triple of edges sharing a vertex) and hence the
quantum factorials in the denominator of the quantum binomials forming the summands in Formula (5) have arguments < n and are nonzero at A = exp 2niπ. By contrast, the numerator has a simple zero exp 2niπ when k ranges in [max({Ti}), n − 2] and a double zero
when k ranges in [n − 1, min({Qj})]. The claim is thus proved.
Our second claim is now that for all k ∈ [max({Ti}), n − 3), the ratio evn Sk+1 Sk is real positive. Indeed: Sk+1 Sk = − {k + 1}{Q (n) 1 − k}{Q (n) 2 − k}{Q (n) 3 − k} {k + 1 − T1(n)}{k + 1 − T2(n)}{k + 1 − T2(n)}{k + 1 − T4(n)}
where we let Q(n)j and Ti(n)be the “squares and triangles” associated to the coloring cn as
in Definition 3.1. Since {k} = 2i sin(πkn ), by the same estimates as in the previous claim the ratio is then positive real as {k + 1} is a negative multiple of 2i. So max{|Sk|} ≤
|evn PkSk)| ≤ (n − 2 − max{Ti})max{|Sk|}, and we are left to find the terms Sk for which
|Sk| is maximal.
Now remark that since the arguments of the factorials in the denominators of Skall belong
to [0, n), their evaluations (by Lemma A.2) grow respectively like ik−Tiexp(−n πΛ(π k−Ti n )) and iQj−kexp(−n πΛ(π Qj−k
n )). By contrast the numerator grows like
ik+2(−1)k+1+n2n2exp −iπ 2n exp −π nΛ πk n ,
so, taking into account the sign (−1)k in front of the multinomial, the kth-summand grows
like: (−1)n2n2exp −iπ 2n {1} exp n π −Λ πk n +X i Λ πk − Ti n +X j Λ πQj− k n . Now let us define numbers τi, νj by
lim
n→∞π
Ti(n)
n = 3π −
γ(e) + γ(e0) + γ(e00)
2 =: 3π − τi 2 lim n→∞π Q(n)j n = 4π −
γ(e) + γ(e0) + γ(e00) + γ(e000)
2 =: 4π −
νj
2 for the edges e, e0, e00, e000 naturally associated to Ti or Qj. Letting πkn = 2π −
s 2 and f (s) = −Λ2π − s 2 +X i Λ −s + τi 2 − π −X j Λ −2π +νj− s 2
and comparing with Equation (16), we see that f (s) = = (U (exp (is))) and that the growth rate of the logarithm of the norm of the evaluation of the sum is given by:
max
s∈[0,min({τi−2π})]
n πf (s).
We thus search for the maximum of f and imposing f0(s) = 0 we get an equation of the form: | sin 2π − s 2 sin ν1−s 2 sin ν2−s 2 sin ν3−s 2 | | sin τ1−s−2π 2 sin τ2−s−2π 2 sin τ3−s−2π 2 sin τ4−s−2π 2 | = 1
Recalling that νj− τi> 0 for all i, j (see Example 3.1), the above equation is equivalent to:
sin 2π − s 2 sin ν1−s 2 sin ν2−s 2 sin ν3−s 2 sin τ1−s−2π 2 sin τ2−s−2π 2 sin τ3−s−2π 2 sin τ4−s−2π 2 = 1
Replacing sin(x) by eix−e2i−ix and setting A := − exp (−iα) (and similarly for all the angles) and z = − exp(is) one gets a polynomial equation of degree 2 in z whose solutions were shown by Murakami and Yano to be z+, z− given in Formula (17). (Even if this is not
apparent from the formula, as soon as the Gram matrix in Equation (18) has negative determinant, then |z−| = |z+| = 1.) In particular, in the case of a regular ideal octahedron
(i.e. a maximally truncated tetrahedron), whose exterior dihedral angles are all π, f (s) = 4Λ s
2 + 4Λ π 2−
s
2 , s ∈ [0, π], whose extremum is attained at s = π
2 and its value is
8Λ π4 > 0. By Theorem A.1 the corresponding point is then z− because Vol(T et) =
= (V (z−)) = −= (V (z+)) = = U (z −)−U (z+) 2 > 0.
One concludes the proof by putting together all the terms in the asymptotical behavior
of Formula (5) and comparing with Theorem A.1.
We are now able to prove Theorem 1.4 in full generality. We do this by induction on the number n of moves of the form producing Γ from a tetrahedron. If n = 0 then Γ is the 1-skeleton of a hyperideal hyperbolic tetrahedron whose possible angles are provided by Example 3.1. For such a graph, Theorem A.3 proves Theorem 1.4. If now Γ can be obtained from Γ0 (for which we assume Theorem 1.4 holds) by a single move, suppose that α, β, γ are the exterior angles on the three edges of Γ0 involved in the move. Observe that any geometric structure of the hyperbolic polyhedron PΓtrunccan be obtained by gluing two polyhedra PΓtrunc0 and T ettrunc, along the face corresponding to the triple of edges of Γ0
where the move is applied. Conversely if PΓtrunc0 and T ettrunc are equipped with geometric
structures such that the dihedral angles along the truncation faces to be matched are the same, then they can be glued to form a hyperideal hyperbolic structure on PΓtrunc. Clearly the volumes add up: Vol(Ptrunc
Γ ) = Vol(P trunc
Γ0 ) + Vol(T ettrunc).
On the quantum side, this follows from equation (8). For any coloring cn on Γ one has
hΓ, cniU= hΓ0, c0ni
UhT et, c00 ni
U
where c0n and c00n are the restrictions of cn to the edges of Γ0 and T et respectively (here we
use the natural injections E(Γ0) → E(Γ) and E(T et) → E(Γ) to restrict the cn). Then one
concludes by induction using Theorem A.3.
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